# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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\u142\ufffdi \u142i and the function w(x) are always positive. Therefore, we have Æi Æ\ufffdi and the eigenvalues Æi are real. For the situation where i 6 j and Æi 6 Æ\ufffdj , the integral in equation (3.21) must vanish, \u142\ufffdj (x)\u142i(x)w(x) dx 0 (3:22) Thus, the set of functions \u142i(x) for non-degenerate eigenvalues are mutually orthogonal when integrated with a weighting function w(x). Eigenfunctions corresponding to degenerate eigenvalues can be made orthogonal as discussed earlier. The discussion above may be generalized to more than one variable. In the general case, equation (3.18) is replaced by A^\u142i(q1, q2, . . .) Æiw(q1, q2, . . .)\u142i(q1, q2, . . .) (3:23) and equation (3.22) by \u142\ufffdj (q1, q2, . . .)\u142i(q1, q2, . . .)w(q1, q2, . . .) dq1 dq2 . . . 0 (3:24) Equation (3.18) can also be transformed into the more usual form, equation (3.5). We first define a set of functions öi(x) as öi(x) [w(x)]1=2\u142i(x) \u142i(x)=u(x) (3:25) where u(x) [w(x)]ÿ1=2 (3:26) The function u(x) is real because w(x) is always positive and u(x) is positive because we take the positive square root. If w(x) approaches infinity at any point within the range of hermiticity of A^ (as x approaches infinity, for example), then \u142i(x) must approach zero such that the ratio öi(x) approaches zero. Equation (3.18) is now multiplied by u(x) and \u142i(x) is replaced by u(x)öi(x) u(x)A^u(x)öi(x) Æiw(x)[u(x)]2öi(x) If we define an operator B^ by the relation B^ u(x)A^u(x) and apply equation (3.26), we obtain B^öi(x) Æiöi(x) which has the form of equation (3.5). We observe that 74 General principles of quantum theory \u142\ufffdj A^\u142i dx ö\ufffdj uA^uöi dx ö\ufffdj B^öi dx (A^\u142 j) \ufffd\u142i dx (A^uö j) \ufffduöi dx (B^ö j) \ufffdöi dx Since A^ is hermitian with respect to the \u142is, the two integrals on the left of each equation equal each other, from which it follows that ö\ufffdj B^öi dx (B^ö j) \ufffdöi dx and B^ is therefore hermitian with respect to the öis. 3.4 Eigenfunction expansions Consider a set of orthonormal eigenfunctions \u142i of a hermitian operator. Any arbitrary function f of the same variables as \u142i defined over the same range of these variables may be expanded in terms of the members of set \u142i f X i ai\u142i (3:27) where the ais are constants. The summation in equation (3.27) converges to the function f if the set of eigenfunctions is complete. By complete we mean that no other function g exists with the property that hg j\u142ii 0 for any value of i, where g and \u142i are functions of the same variables and are defined over the same variable range. As a general rule, the eigenfunctions of a hermitian operator are not only orthogonal, but are also complete. A mathematical criterion for completeness is presented at the end of this section. The coefficients ai are evaluated by multiplying (3.27) by the complex conjugate \u142\ufffdj of one of the eigenfunctions, integrating over the range of the variables, and noting that the \u142is are orthonormal h\u142 j j f i \u142 j \ufffd\ufffd\ufffd\ufffdX i ai\u142i * + X i aih\u142 j j\u142ii a j Replacing the dummy index j by i, we have ai h\u142i j f i (3:28) Substitution of equation (3.28) back into (3.27) gives f X i h\u142i j f i\u142i (3:29) 3.4 Eigenfunction expansions 75 Completeness We now evaluate h f j f i in which f and f \ufffd are expanded as in equation (3.27), with the two independent summations given different dummy indices h f j f i X j a j\u142 j \ufffd\ufffd\ufffd\ufffdX i ai\u142i * + X j X i a\ufffdj a jh\u142 j j\u142ii X i jaij2 Without loss of generality we may assume that the function f is normalized, so that h f j f i 1 and X i jaij2 1 (3:30) Equation (3.30) may be used as a criterion for completeness. If an eigenfunc- tion \u142n with a non-vanishing coefficient an were missing from the summation in equation (3.27), then the series would still converge, but it would be incomplete and would therefore not converge to f . The corresponding coeffi- cient an would be missing from the left-hand side of equation (3.30). Since each term in the summation in equation (3.30) is positive, the sum without an would be less than unity. Only if the expansion set \u142i in equation (3.27) is complete will (3.30) be satisfied. The completeness criterion can also be expressed in another form. For this purpose we need to introduce the variables explicitly. For simplicity we assume first that f is a function of only one variable x. In this case, equation (3.29) is f (x) X i \u142\ufffdi (x9) f (x9) dx9 \ufffd \ufffd \u142i(x) where x9 is the dummy variable of integration. Interchanging the order of summation and integration gives f (x) X i \u142\ufffdi (x9)\u142i(x) " # f (x9) dx9 Thus, the summation is equal to the Dirac delta function (see Appendix C)X i \u142\ufffdi (x9)\u142i(x) ä(xÿ x9) (3:31) This expression, known as the completeness relation and sometimes as the closure relation, is valid only if the set of eigenfunctions is complete, and may be used as a mathematical test for completeness. Notice that the completeness relation (3.31) is not related to the choice of the arbitrary function f , whereas the criterion (3.30) is related. The completeness relation for the multi-variable case is slightly more complex. When expressed explicitly in terms of its variables, equation (3.29) is 76 General principles of quantum theory f (q1, q2, . . .) X i \u142\ufffdi (q91, q92, . . .) f (q91, q92, . . .)w(q91, q92, . . .) dq91, dq92, . . . \ufffd \ufffd 3 \u142i(q1, q2, . . .) Interchanging the order of summation and integration gives f (q1, q2, . . .) X i \u142\ufffdi (q91, q92, . . .)\u142i(q1, q2, . . .) " # 3 f (q91, q92, . . .)w(q91, q92, . . .) dq1 dq2 . . . so that the completeness relation takes the form w(q91, q92, . . .) X i \u142\ufffdi (q91, q92, . . .)\u142i(q1, q2, . . .) ä(q1 ÿ q91)ä(q2 ÿ q92) . . . (3:32) 3.5 Simultaneous eigenfunctions Suppose the members of a complete set of functions \u142i are simultaneously eigenfunctions of two hermitian operators A^ and B^ with eigenvalues Æi and âi, respectively A^\u142i Æi\u142i B^\u142i âi\u142i If we operate on the first eigenvalue equation with B^ and on the second with A^, we obtain B^A^\u142i ÆiB^\u142i Æiâi\u142i A^B^\u142i âiA^\u142i Æiâi\u142i from which it follows that (A^B^ÿ B^A^)\u142i [A, B]\u142i 0 Thus, the functions \u142i are eigenfunctions of the commutator [A^, B^] with eigenvalues equal to zero. An operator that gives zero when applied to any member of a complete set of functions is itself zero, so that A^ and B^ commute. We have just shown that if the operators A^ and B^ have a complete set of simultaneous eigenfunctions, then A^ and B^ commute. We now prove the converse, namely, that eigenfunctions of commuting operators can always be constructed to be simultaneous eigenfunctions. Suppose that A^\u142i Æi\u142i and that [A^, B^] 0. Since A^ and B^ commute, we have 3.5 Simultaneous eigenfunctions 77 A^B^\u142i B^A^\u142i B^(Æi\u142i) ÆiB^\u142i Therefore, the function B^\u142i is an eigenfunction of A^ with eigenvalue Æi. There are now two possibilities; the eigenvalue Æi of A^ is either non- degenerate or degenerate. If Æi is non-degenerate, then it corresponds to only one independent eigenfunction \u142i, so that the function B^\u142i is proportional to \u142i B^\u142i âi\u142i where âi is the proportionality constant and therefore the eigenvalue of B^ corresponding to \u142i. Thus, the function \u142i is a simultaneous eigenfunction of both A^ and B^. On the other hand, suppose the eigenvalue Æi is degenerate. For simplicity, we consider the case of a doubly degenerate eigenvalue Æi; the extension to n- fold degeneracy is straightforward. The function \u142i is then any linear combina- tion of two linearly independent, orthonormal eigenfunctions \u142i1 and \u142i2 of A^ corresponding to the eigenvalue Æi \u142i c1\u142i1 c2\u142i2 We